Question: What is the greatest common factor of $15n^{5}$, $30n^{3}$, and $45n^{2}$ ?
Explanation: Let's factor each monomial to its prime factors: $\begin{aligned} 15n^{5}&=(3)(5)(n)(n)(n)(n)(n) \\\\ 30n^{3}&=(2)(3)(5)(n)(n)(n) \\\\ 45n^{2}&=(3)(3)(5)(n)(n) \end{aligned}$ We want the largest set of factors that's included in all three monomials. All of the monomials have one factor of $ 3$, one factor of $ 5$, and two factors of $ n$ : $\begin{aligned} 15n^{5}&=( 3)( 5)( n)( n)(n)(n)(n) \\\\ 30n^{3}&=(2)( 3)( 5)( n)( n)(n) \\\\ 45n^{2}&=( 3)(3)( 5)( n)( n) \end{aligned}$ This is the greatest common factor: $( 3)( 5)( n)( n)=15n^2$